Regge amplitudes from AdS/CFT - Jagiellonian Universityth- › school › 2011 › talks ›...

Regge amplitudes from AdS/CFT

Robi Peschanski

Institut de Physique Theorique, Saclay

Zakopane Lectures (May 2011)

Initial work: R.Janik, R.P., hep-th//9907177,0003059,/01100241rst Lecture (1-4): M.Giordano, S.Seki, R.P., arXiv:1106.xxxx [hep-th]2nd Lecture (5-8): M.Giordano, R.P., arXiv:1105.xxxx [hep-th]

Robi Peschanski (Saclay) Regge amplitudes from AdS/CFT 1 / 19

Motivation

Aim: Use the AdS/CFT and Gauge/Gravity duality to study high-energy softscattering amplitudes.

String Theory Ancestor: The “dual” Regge amplitudes for Strong Interactions:

Shapiro-Virasoro AmplitudeVeneziano Amplitude

{q^2

A (s,q^2) A (s,q^2)R P

s{

AR,P(s, t = −q2)?−→

∑

i

(

s

s0

)αi (t)

βi (t)


Questions

Old string scenario (1968-1974): Problems

Consistent (super) string: (10) 26 dimensions and Zero-mass on-shell states

Reggeon exchange: → Gauge interactions

Pomeron exchange: → Gravity

New string scenario (1998-2011-): AdS/CFT correspondence andGauge/Gravity duality at strong coupling

Question # 1: Regge behaviour for the conformal N = 4 gauge field theoryfrom AdS/CFT?

Question # 2: Regge behaviour for a non conformal gauge field theory ∼QCD from Gauge/Gravity duality?


Outline of the lectures

1 Introduction: old/new Reggeon problem

2 AdS/CFT and minimal surfaces

3 gg amplitude in N = 4 sYM theory

4 Regge Amplitudes in N = 4 sYM theory


The AdS/CFT correspondence

N = 4 Super Yang-Mills theory ≡ Superstrings on AdS5 × S5

strong coupling (semi-)classical stringsnonperturbative physics or supergravity

very difficult ‘easy’weak coupling highly quantum regime

‘easy’ very difficult

New ways of looking at nonperturbative gauge theory physics...

Intricate links with General Relativity...

This is an equivalence! Any state/phenomenon on the gauge theory sideshould have its dual counterpart...

Caveat: the dual counterpart does not neccessarily have to be in the wellunderstood (super)gravity sector...


The AdS/CFT correspondence

Maldacena (1997):

Weak Gravity

Strong Gravity

Strong CouplingWeak Coupling

5+1

3+1

5+1

3+1

Macroscopic

AdS CFT

Microscopic}

Gravity Source

}

N-Branes

55AdS x S Superstring SU(N) Gauge Theory on the N-Branes

g N -> 0( (

S S

-1g N -> 0

Duality


Gauge/Gravity Duality

Gauge/Gravity Duality: a wider concept

Schomerus

Open ⇔ Closed String duality

Tree-level Closed String ⇔ 1-loop Open String

Classical Gravity ⇔ Quantum Gauge field Theory

Small/Large Distance ⇒ Weak/Strong Gauge coupling ?


Holography

G/G Tool: Wilson Loops ⇔ Minimal Surfaces

HORIZON

Boundary

〈e iPR

C~A·~dl〉 =

∫

Σ

e−Area(Σ)

α′ ≈ e−Min.Area

α′ ×Fluctuations

Confining theory ∼ QCD N = 4 Conformal Theory

Linear potential A ∝ T × L Coulombic potential A ∝ T/L

Quasi-planar Approximation Non-planar minimal surfaces∼ QCD Regge amplitude ? N = 4 sYM Regge amplitude?


Gauge/Gravity Method

G/G Tool: Eikonal approximation ⇔ Helicoidal GeometryJanik, R.P. 99-01

L

θ

t

x

y

T

T

a(L = |~b|, χ) ≡ i

2s

∫

d2~q

(2π)2e−i~q·~b AR(s, t)

⇔ a(L, θ→−iχ, T ) = Z−1

∫

DC 〈W [C]〉 ∼ e−1

α′AHelicoid

Confining theory ∼ QCD N = 4 Conformal Theory

∼ R4 Helicoid known AdS5 “Helicoid” unknownAnalytic Continuation → Minkowski ? N = 4 sYM Regge amplitude?


Gauge/Gravity Method

Flat space example: Eikonal approximation ⇔ Helicoidal Geometry

Area = LT

√

1 +T 2θ2

L2+

L2

2θlog

√

1 + T 2θ2

L2 + θ TL

√

1 + T 2θ2

L2 − θ TL


Alday-Maldacena Amplitude

4− gluon amplitude: ⇔ Minkowskian Minimal surface (z near ∞)Alday, Maldacena (2007)

Y1

Y2

Y0

Y1

Y2

• Minkowskian Boundary: Polygon of light-like gluons (+boosts)

• Exact minimal surface in T-dual AdS5 metric: ds2 = R2

r2

(

ηµνdyµdyν + dr2)

• Amplitude from minimal Area:

A(s, t) = e−iA = exp

[

2iSdiv(s) + 2iSdiv(t) +

√λ

8π

(

logs

t

)2

+ C

]

iSdiv(p) = − 1

ǫ2

1

2π

√

λµ2ǫ

(−p)ǫ− 1

ǫ

1

4π(1 − log 2)

√

λµ2ǫ

(−p)ǫ, (p = s, t)


Alday-Maldacena Amplitude

Regge limit: (−s) → ∞, (−t) fixed

• Divergent part

iSdiv(p) = − 1

ǫ2

√λ

2π+

1

ǫ

√λ

4πlog

−2p

eµ2−

√λ

16πlog2 −p

µ2+

√λ

8π(1 − log 2) log

−p

µ2

• Regge Amplitude:

A(s, t) = Adiv · AR · eO(ǫ)

Adiv(s, t) = exp

[

− 1

ǫ2

2√

λ

π+

1

ǫ

√λ

2π

(

log−s

µ2+ log

−t

µ2− 2(1 − log 2)

)]

AR(s, t) =

(−s

µ2

)− 14 f (λ) log −t

µ2 +g(λ)

exp

(

g(λ)

4log

−t

µ2+ C

)

• Regge Trajectory:

αR(t) = − f (λ)4 log −t

µ2 + g(λ)4 f (λ) =

√λ

π; g(λ) =

√λ

π(1 − log 2)


AdS5 Minimal Surfaces

A show of AdS5 Minimal surfaces


AdS5 Minimal surfaces

Regge limit: Minimal Surface at the IR z →∞



Regge limit: Minimal Surface at the UV z →0



Regge limit: Wilson Lines Boundary z = 0

Parallels gg double helix qq double helix

Different “helicoids” in AdS5 for gg and qq !!! So what to do?


qq Regge Amplitude in N = 4 sYM theory

Minimal Surface for qq amplitude: Conformal Transform of HelicoidGiordano,R.P.,Seki (2011)

• Invariance by 5-d Inversion

xµ → xµ

|xµ|2 + z2, z → z

|xµ|2 + z2

• Euclidean Two-cusp Dominance

Aquarkminimal = 2ΓE

cusp(θ) log(Mqb) + · · ·Robi Peschanski (Saclay) Regge amplitudes from AdS/CFT 17 / 19

qq Regge Amplitude in N = 4 sYM theory

Analytic Continuation Euclid → Minkowski

• Cusp Minimal Surface in AdS5

Kruczenski (2002)

Y0

r

Y1

• Cusp: θ = π + iχ, Γcusp(χ) → − f (λ)4 χ for χ ≫ 1

Aquarkminimal(Mb, χ) = − f (λ)

2log

Mb

χ+ Ψ(χ)

• Back to Momentum space:

AquarkRegge(s, t) ≈ 2(−s/M2)

− f4 log −t

µ2 + f2 log 2


qq vs. gg Regge amplitudes

Results: Eikonal approximation vs. Alday-Maldacena

qq 2−cusp amplitude gg 4 → 2−cusp amplitude

L

θ

t

x

y

T

T

(a) (b)

1

2 2

1t

ss

t

αR(t) = − f (λ)4 log −t

µqq2 αR(t) = − f (λ)

4 log −tµgg

2


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